Find the number of positive integers that satisfy both the following conditions:

- Each digit is a 1 or a 2 or a 3

- The sum of the digits is 5

kittykat Aug 16, 2024

#1**+1 **

We could probably use casework to solve this problem.

The first case to look for is only two digit numbers.

We have 23 and 32, so there are 2 possibilities.

Next, consider 3 digit numbers.

We have 311, 131, 113 for 3 more.

We also have 221, 212, and 122 to add 3 more on.

Next, consider 4 digit numbers.

The only numbers that work are numbers with three 1s and one 2. There are 4 ways we could do this.

Lastly, we have 5 digit numbers. The only one we have to consider is \(111111\), one possibility.

Now, we add up the numbers. We get \(2+3+3+4+1 = 13.\)

So 13 is our answer!

Thanks! :)

NotThatSmart Aug 16, 2024